
Journal of Lie Theory 18 (2008), No. 1, 215241 Copyright Heldermann Verlag 2008 Borel Subalgebras of RootReductive Lie Algebras Elizabeth DanCohen Dept. of Mathematics, University of California, Berkeley, CA 94720, U.S.A. edc@math.berkeley.edu [Abstractpdf] \def\g{{\frak g}} \def\l{{\frak l}} \def\o{{\frak o}} \def\s{{\frak s}} This paper generalizes the classification of I. Dimitrov and I. Penkov [{\it Borel subalgebras of $\g\l(\infty)$}, Resenhas 6 (2004) 153163] of Borel subalgebras of $\g\l_\infty$. Rootreductive Lie algebras are direct limits of finitedimensional reductive Lie algebras along inclusions preserving the root spaces with respect to nested Cartan subalgebras. A Borel subalgebra of a rootreductive Lie algebra is by definition a maximal locally solvable subalgebra. The main general result of this paper is that a Borel subalgebra of an infinitedimensional indecomposable rootreductive Lie algebra is the simultaneous stabilizer of a certain type of generalized flag in each of the standard representations. \par For the three infinitedimensional simple rootreductive Lie algebras more precise results are obtained. The map sending a maximal closed (isotropic) generalized flag in the standard representation to its stabilizer hits Borel subalgebras, yielding a bijection in the cases of $\s\l_\infty$ and $\s\p_\infty$; in the case of $\s\o_\infty$ the fibers are of size one and two. A description is given of a nice class of toral subalgebras contained in any Borel subalgebra. Finally, certain Borel subalgebras of a general rootreductive Lie algebra are seen to correspond bijectively with Borel subalgebras of the commutator subalgebra, which are understood in terms of the special cases. Keywords: Locally finite Lie algebra, rootreductive Lie algebra, Borel subalgebra, maximal locally solvable subalgebra. MSC: 17B65, 17B30, 17B05 [ Fulltextpdf (286 KB)] for subscribers only. 